Newform orbit 89.2.a.a

$q$-expansion

$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - 4q^{7} + 3q^{8} - 2q^{9} + q^{10} - 2q^{11} + q^{12} + 2q^{13} + 4q^{14} + q^{15} - q^{16} + 3q^{17} + 2q^{18} - 5q^{19} + q^{20} + 4q^{21} + 2q^{22} + 7q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$. The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 2$

Atkin–Lehner signs

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$. The Fricke sign is $w_N = +1$, so the sign of the functional equation is $(-1)^{k/2} w_N = -1$: each L-function in the orbit has odd analytic rank.

Elliptic curve

By modularity this newform is the one attached to the isogeny class 89.a (Cremona's 89a). Its optimal curve 89a1 is

$y^2 + xy + y = x^3 + x^2 - x$

of rank 1 with torsion of order 1. Its $a_p$, computed by Sagebrush's point-counting engine, equal the newform's for every prime $p < 1000$.

Traces of Hecke eigenvalues

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{1/2}$ for the primes $p < 1000$ not dividing 89, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$:

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.163 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

newform_orbits(89, 2)[0]   # 89.2.a.a

cargo run --release -p sagebrush-web --example atlas -- 89:2 1000